Triangle-of-Means Recurrence (Math Spec)

Purpose

Define the mathematical core used for state updates.

Definitions

• Projection: \(X_t = W_x H_{t-1} + b_x\)
• Reception: \(Y_t = W_y I_t + b_y\)
• Base (error): \(b_t = (Y_t - X_t)/2\)
• Neutral (geometric mean): \(a_t = \exp(0.5(\ln(|X_t|+\epsilon)+\ln(|Y_t|+\epsilon)))\)
• Carrier (arithmetic mean): \(c_t = (Y_t + X_t)/2\)

Rotation / Phase

• Phase increment: \(\Delta \phi_t \in \{5,7,11,13\}\times 2\pi/24\), or learned variant.
• Rotation operator: define block-rotations or unitary update here.

Update Rule

State proposal: \(z_t = f([b_t,a_t,c_t])\). Final: \(H_t = g( R_{\Delta \phi_t}(z_t), H_{t-1})\).

Coherence Objective

\(\mathcal{L}_{coh} = \mathbb{E}[gate(b_t)\cdot (1-\cos(H_{t-1},H_t))]\)